Parallel calculation of a linear mapping on a computer network

Abstract We are interested in the parallel computation of a linear mapping of n real variables by a network of computers with restricted means of communication between them and without any common memory. Let M n×n ( R ) denote the algebra of n × n real matrices, and let G be the graph associated with a binary, reflexive and symmetric relation R over {1,2, …, n }. We define A R = {AϵM n×n ( R ):a ij ≠ 0 implies iRj} A matrix M∈M n×n ( R ) is said to be realizable on G if it can be expressed as a product of elements of A R . Therefore, every matrix of M n×n ( R ) is realizable on G if and only if A R generates M n×n ( R ) . We show that A R generates M n×n ( R ) if and only if G is connected.